Algebra Know-It-ALL

If you have a high-end calculator that goes to two or three dozen digits such as the sort found in some

computers, you can leave all the displayed digits in the figures as you go through the arithmetic. Using

the memory store and recall functions, those extra digits can be kept in the process, and that will keep the

rounding error extremely small. When you get the final answer and round it off to, say, six decimal places

instead of the two or three dozen, the error won’t show up. You can play around with a good calculator to

see how this works.

Are you still confused?

You might also ask, “What about the logarithm of 0, or of any negative number?” The answer is, “These

are not defined in the set of real numbers, no matter what the base.” To understand why, check out what

happens if you try to calculate the common log of −2. Suppose that

log 10 − 2 =x

This can be rewritten as

10 x=− 2

What’s the value of x? That’s hard to say, but it’s not a real number! No matter what real number you

choose for x, the value of 10x is positive. If you change −2 to any other negative number, or to 0, you run

into the same problem. It’s impossible to find any real number x, such that 10x is not a positive real.

Here’s a challenge!

Find the natural log of 238.967 from its common log using the above conversion formula. Express your

answer to three decimal places. Then take the natural log of 238.967 directly with your calculator, round

it off to three decimal places, and compare that result with the first one.

Solution

Working with a calculator, you should get

log 10 238.967 ≈ 2.378

Now remember the conversion formula:

lnx≈ 2.302588 log 10 x

In this example, you should multiply 2.302588 by 2.378. When you round the answer off to three decimal

places, you should get

2.302588× 2.378 ≈ 5.476

When you take the natural log of 238.967 directly with a calculator and then round off to three decimal

places, you’ll see that

ln 238.967 ≈ 5.476

486 Logarithms and Exponentials